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a(n) = 4* A117384(n) - n; a self-inverse permutation of the natural numbers.
+20
2
3, 6, 1, 8, 11, 2, 13, 4, 15, 18, 5, 20, 7, 22, 9, 24, 27, 10, 29, 12, 31, 14, 33, 16, 35, 38, 17, 40, 19, 42, 21, 44, 23, 46, 25, 48, 51, 26, 53, 28, 55, 30, 57, 32, 59, 34, 61, 36, 63, 66, 37, 68, 39, 70, 41, 72, 43, 74, 45, 76, 47, 78, 49, 80, 83, 50, 85, 52, 87, 54, 89, 56
Connell sequence: 1 odd, 2 even, 3 odd, ...
(Formerly M0962 N0359)
+10
38
1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, 19, 21, 23, 25, 26, 28, 30, 32, 34, 36, 37, 39, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 122
COMMENTS
a(t_n) = a(n(n+1)/2) = n^2 relates squares to triangular numbers. - Daniel Forgues
The natural numbers not included are A118011(n) = 4n - a(n) as n=1,2,3,... - Paul D. Hanna, Apr 10 2006 As a triangle with row sums = A069778 (1, 6, 21, 52, 105, ...): /Q 1;/Q 2, 4;/Q 5, 7, 9;/Q 10, 12, 14, 16;/Q ... . - Gary W. Adamson, Sep 01 2008 The triangle sums, see A180662 for their definitions, link the Connell sequence A001614 as a triangle with six sequences, see the crossrefs. - Johannes W. Meijer, May 20 2011
REFERENCES
C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 276.
C. A. Pickover, The Mathematics of Oz, Chapter 39, Camb. Univ. Press UK 2002.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Ian Connell and Andrew Korsak, Problem E1382, Amer. Math. Monthly, 67 (1960), 380.
FORMULA
a(n) = 2*n - floor( (1+ sqrt(8*n-7))/2 ).
a(1) = 1; then a(n) = a(n-1)+1 if a(n-1) is a square, a(n) = a(n-1)+2 otherwise. For example, a(21)=36 is a square therefore a(22)=36+1=37 which is not a square so a(23)=37+2=39 ... - Benoit Cloitre, Feb 07 2007 Let the sequence be written in the form of the triangle in the EXAMPLE section below and let a(n) and a(n+1) belong to the same row of the triangle. Then a(n)*a(n+1) + 1 = a( A000217( A118011(n))) = A000290( A118011(n)). - Ivan N. Ianakiev, Aug 16 2013 G.f. 2*x/(1-x)^2 - (x/(1-x))*sum(n>=0, x^(n*(n+1)/2))
= 2*x/(1-x)^2 - (Theta2(0,x^(1/2)))*x^(7/8)/(2*(1-x)) where Theta2 is a Jacobi theta function.
a(n) = 2*n-1 - Sum(i=0..n-2, A023531(i)). (End)
EXAMPLE
Written as a triangle the sequence begins:
1;
2, 4;
5, 7, 9;
10, 12, 14, 16;
17, 19, 21, 23, 25;
26, 28, 30, 32, 34, 36;
37, 39, 41, 43, 45, 47, 49;
50, 52, 54, 56, 58, 60, 62, 64;
65, 67, 69, 71, 73, 75, 77, 79, 81;
82, 84, 86, 88, 90, 92, 94, 96, 98, 100;
...
Right border gives A000290, n >= 1. (End)
MATHEMATICA
lst={}; i=0; For[j=1, j<=4!, a=i+1; b=j; k=0; For[i=a, i<=9!, k++; AppendTo[lst, i]; If[k>=b, Break[]]; i=i+2]; j++ ]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *) row[n_] := 2*Range[n+1]+n^2-1; Table[row[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Oct 25 2013 *)
PROG
(Haskell)
a001614 n = a001614_list !! (n-1)
a001614_list = f 0 0 a057211_list where
f c z (x:xs) = z' : f x z' xs where z' = z + 1 + 0 ^ abs (x - c)
(Python)
from math import isqrt
def A001614(n): return (m:=n<<1)-(k:=isqrt(m))-int((m<<2)>(k<<2)*(k+1)+1) # Chai Wah Wu, Jul 26 2022
CROSSREFS
Triangle columns: A002522, A117950 (n>=1), A117951 (n>=2), A117619 (n>=3), A154533 (n>=5), A000290 (n>=1), A008865 (n>=2), A028347 (n>=3), A028878 (n>=1), A028884 (n>=2), A054569 [T(2*n,n)]. Triangle sums (see the comments): A069778 (Row1), A190716 (Row2), A058187 (Related to Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Fi1, Fi2, Ze1 and Ze2), A000292 (Related to Kn3, Kn4, Ca3, Ca4, Gi3 and Gi4), A190717 (Related to Ca1, Ca2, Ze3, Ze4), A190718 (Related to Gi1 and Gi2). (End)
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Mar 16 2001
Complement of the Connell sequence ( A001614); a(n) = 4*n - A001614(n).
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7
3, 6, 8, 11, 13, 15, 18, 20, 22, 24, 27, 29, 31, 33, 35, 38, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 61, 63, 66, 68, 70, 72, 74, 76, 78, 80, 83, 85, 87, 89, 91, 93, 95, 97, 99, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 123, 125, 127, 129, 131, 133, 135, 137, 139
COMMENTS
a(n) is the position of the second appearance of n in A117384, where A117384(m) = A117384(k) and k = 4* A117384(m) - m. The Connell sequence ( A001614) is generated as: 1 odd, 2 even, 3 odd, ...
FORMULA
a(n) = 2*n + 1 + Sum(j=0 .. n-2, A023531(j)). G.f. 2*x/(1-x)^2 + x/(1-x) * Sum(j=0..infinity, x^(j*(j+1)/2))
= 2*x/(1-x)^2 + x^(7/8)/(2-2*x) * Theta2(0,sqrt(x)), where Theta2 is a Jacobi theta function. (End)
PROG
(Python)
from math import isqrt
def A118011(n): return (m:=n<<1)+(k:=isqrt(m))+int((m<<2)>(k<<2)*(k+1)+1) # Chai Wah Wu, Jul 26 2022
Positive integers, each occurring twice in the sequence, such that a(n) = a(k) when n+k = 5*a(n), starting with a(1)=1 and filling the next vacant position with the smallest unused number.
+10
3
1, 2, 3, 1, 4, 5, 6, 2, 7, 8, 9, 3, 10, 11, 4, 12, 13, 14, 5, 15, 16, 17, 6, 18, 19, 7, 20, 21, 22, 8, 23, 24, 25, 9, 26, 27, 10, 28, 29, 30, 11, 31, 32, 12, 33, 34, 35, 13, 36, 37, 38, 14, 39, 40, 15, 41, 42, 43, 16, 44, 45, 46, 17, 47, 48, 18, 49, 50, 51, 19, 52, 53, 20, 54, 55, 56
FORMULA
a(5*a(n)-n) = a(n). Conjecture: Lim_inf a(n)/n = (5-sqrt(5))/10; Lim_sup a(n)/n = (5+sqrt(5))/10.
PROG
(PARI) {a(n)=local(A=vector(n), m=1); for(k=1, n, if(A[k]==0, A[k]=m; if(5*m-k<=#A, A[5*m-k]=m); m+=1)); A[n]}
Positive integers, each occurring twice in the sequence, such that a(n) = a(k) when n+k = 6*a(n), starting with a(1)=1 and filling the next vacant position with the smallest unused number.
+10
2
1, 2, 3, 4, 1, 5, 6, 7, 8, 2, 9, 10, 11, 12, 3, 13, 14, 15, 16, 4, 17, 18, 19, 5, 20, 21, 22, 23, 6, 24, 25, 26, 27, 7, 28, 29, 30, 31, 8, 32, 33, 34, 9, 35, 36, 37, 38, 10, 39, 40, 41, 42, 11, 43, 44, 45, 46, 12, 47, 48, 49, 13, 50, 51, 52, 53, 14, 54, 55, 56, 57, 15, 58, 59, 60, 61
FORMULA
a(6*a(n)-n) = a(n). Conjecture: Lim_inf a(n)/n = (3-sqrt(3))/6; Lim_sup a(n)/n = (3+sqrt(3))/6.
PROG
(PARI) {a(n)=local(A=vector(n), m=1); for(k=1, n, if(A[k]==0, A[k]=m; if(6*m-k<=#A, A[6*m-k]=m); m+=1)); A[n]}
Primes, each occurring twice, such that a(C(n)) = a(4*n-C(n)) = prime(n), where C is the Connell sequence ( A001614).
+10
1
2, 3, 2, 5, 7, 3, 11, 5, 13, 17, 7, 19, 11, 23, 13, 29, 31, 17, 37, 19, 41, 23, 43, 29, 47, 53, 31, 59, 37, 61, 41, 67, 43, 71, 47, 73, 79, 53, 83, 59, 89, 61, 97, 67, 101, 71, 103, 73, 107, 109, 79, 113, 83, 127, 89, 131, 97, 137, 101, 139, 103, 149, 107, 151
COMMENTS
Terms can be arranged in an irregular triangle read by rows in which row r is a permutation P of the primes in the interval [prime(s), prime(s+rlen-1)], where s = 1+(r-1)*(r-2)/2, rlen = 2*r-1 = A005408(r-1) and r >= 1 (see example). P is the alternating (first term > second term < third term > fourth term < ...) permutation m -> 1, 1 -> 2, m+1 -> 3, 2 -> 4, m+2 -> 5, 3 -> 6, ..., rlen -> rlen where m = ceiling(rlen/2).
The triangle has the following properties.
Row lengths are the positive odd numbers ( A005408). Each even column is equal to the column preceding it.
Row records ( A011756) are in the right border. Indices of row records are the positive terms of A000290. Each row r contains r terms that are duplicated in the next row.
In each row, the sum of terms which are not already listed in the sequence give A007468.
EXAMPLE
Written as an irregular triangle the sequence begins:
2;
3, 2, 5;
7, 3, 11, 5, 13;
17, 7, 19, 11, 23, 13, 29;
31, 17, 37, 19, 41, 23, 43, 29, 47;
53, 31, 59, 37, 61, 41, 67, 43, 71, 47, 73;
79, 53, 83, 59, 89, 61, 97, 67, 101, 71, 103, 73, 107;
...
The triangle can be arranged as shown below so that, in every row, each odd position term is equal to the term immediately below it.
2
3 2 5
7 3 11 5 13
17 7 19 11 23 13 29
31 17 37 19 41 23 43 29 47
...
MATHEMATICA
nterms=64; a=ConstantArray[0, nterms]; For[n=1; p=1, n<=nterms, n++, If[a[[n]]==0, a[[n]]=Prime[p]; If[(d=4p-n)<=nterms, a[[d]]=a[[n]]]; p++]]; a
(* Second program, triangle rows *)
nrows=8; Table[rlen=2r-1; Permute[Prime[Range[s=1+(r-1)(r-2)/2, s+rlen-1]], Join[Range[2, rlen, 2], Range[1, rlen, 2]]], {r, nrows}]
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